Problem: Determine how many solutions exist for the system of equations. ${8x+2y = 18}$ ${12x+3y = 6}$
Answer: Convert both equations to slope-intercept form: ${8x+2y = 18}$ $8x{-8x} + 2y = 18{-8x}$ $2y = 18-8x$ $y = 9-4x$ ${y = -4x+9}$ ${12x+3y = 6}$ $12x{-12x} + 3y = 6{-12x}$ $3y = 6-12x$ $y = 2-4x$ ${y = -4x+2}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x+9}$ ${y = -4x+2}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.